Algebra

Local finite group theory has been an important topic, particularly via the Classification of Finite Simple Groups. Fusion systems are categories defined on p-groups, which have helped simplify and extend much of the theory in the last 20 years and are starting to link group theory, representation theory, and algebraic topology. By Alperin-Goldschmidt's fusion theorem, a saturated fusion system can be completely determined by its essential subgroups. In this talk, I will discuss fusion systems with two-generator essential subgroups and present some other results on fusion systems.

This talk explores the interplay between algebraic structures known as skew braces and set-theoretic solutions to the Yang–Baxter equation (YBE), a central object in mathematical physics and quantum group theory. Skew braces provide a powerful framework for constructing and analysing solutions.

We focus on involutive solutions, in particular on their permutation skew braces that allow for explicit computations and structural analysis.

We introduce the necessary background on skew braces and the YBE, we study the role of the permutation skew brace, which captures essential symmetries of solutions and serves as an effective invariant. As a concrete application, we present a full classification, up to isomorphism, of all involutive solutions of size $p^2$, where $p$ is a prime.

Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, Sergey Fomin and Linus Setiabrata have recently introduced Heronian friezes - the Euclidean analogues of Coxeter’s frieze patterns.  I will explain the way these friezes arise from a polygon in the complex plane, and that a sufficiently generic Heronian frieze is uniquely determined by a small proportion of its entries. Then, I shall proceed to some of my results, that hold in the cyclic case, i.e.  when the vertices of the polygon are placed on the circle. Namely, I will speak about vanishing of certain determinants in the cyclic case, as well as explain some interesting algebraic relations that hold between the entries of the cyclic Heronian frieze.

This talk will not assume any knowledge of derived categories.
Two rings are "Morita equivalent" if they have equivalent module categories, and
if a property of rings depends only on the module category, then it is called
"Morita invariant". More generally, two rings are "derived equivalent" if they
have equivalent derived categories, and if a property of rings depends only on
the derived category, then it is called "derived invariant".
Fairly recently, Manuel Saorin asked me if I knew whether right coherence was a
derived invariant property. I didn't, but when my long term memory kicked in, I
realised that an example hidden in my 36 year old PhD thesis could be used to
give a counterexample.
Most of the talk will be an introduction to the background to this question and
variants, but it will finish with some fun examples of rings with strange
properties.

Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.

A large class of algebras (those admitting quasi-Cartan subalgebras) can be reconstructed from naturally occurring inverse semigroups and groupoids. I will describe my work with collaborators comparing constructions of groupoids from inverse semigroups that supported this reconstruction. I will not assume any background on inverse semigroups or groupoids.

We all know that quandles are great, but there seems to have been resistance to their use as knot invariants because they're too hard to work with.  In this talk I will explain an older result of mine with Sheila Miller, giving rigorous mathematical substance to this sociological observation: we show that, in the framework of "Borel reducibility", the isomorphism problem for quandles is as complicated as it could possibly be.  The proof boils down to a very hands-on, combinatorial construction of a quandle from any given graph.

Exceptional sequences were first introduced in triangulated categories by the Moscow school of algebraic geometry. Later, Crawley-Boevey and Ringel studied exceptional sequences in the module categories of hereditary finite-dimensional algebras. Motivated by tau-tilting theory introduced by Adachi, Iyama, and Reiten, Jasso’s reduction for tau-tilting modules, and signed exceptional sequences introduced by Igusa and Todorov, Buan and Marsh developed the theory of (signed) tau-exceptional sequences – a natural generalization of (signed) exceptional sequences that behave well over arbitrary finite-dimensional algebras.

In this talk, we will study (signed) tau-exceptional sequences over the algebra Λ=RQ, where R is a finite-dimensional local commutative algebra over an algebraically closed field, and Q is an acyclic quiver. I will explain how (signed) tau-exceptional sequences over Λ can be fully understood in terms of (signed) exceptional sequences over kQ.

Over an algebraically closed field, Gabriel’s theorem states that the path algebra kQ of a connected quiver is representation-finite if and only if the underlying graph of Q is an ADE Dynkin diagram. Equivalently, kQ is representation-finite precisely when the preprojective algebra of Q is finite-dimensional.

d-Representation-finite (d-RF) algebras, introduced by Iyama and Oppermann, are a generalisation of representation-finite path algebras. Attached to each d-RF algebra is a (d+1)-preprojective algebra. Grant showed that a d-RF algebra is fractional Calabi-Yau if and only if the Nakayama automorphism of its (d+1)-preprojective algebra has finite order.

In this talk, we present a family of algebras which arise from the well-studied 3-preprojective algebras of type A by “taking orbifolds”. We show that a subset of these are themselves 3-preprojective algebras (of type D). Thus we provide new examples of 2-RF algebras, which we show are also fractional Calabi-Yau.

I shall explain Theorem (127) which determines the unique bijection BETWEEN the monomial basis called $(A_n)$- FC elements  AND the set of non-crossing diagrams of $n+1$ strings OF  our well-known Temperley-Lieb algebra, that respects the  diagrammatic multiplication by concatenation along with the two algorithms implementing this bijection and its inverse, in other terms : "Drawing" a basis element into a diagram and "writing" a diagram as member in the monomial basis.

Naturally  we shall find ourselves in CataLand, so I will try to give -as much as our time allows me- some consequences and open problems coming from the above holly marriage, that is to explain why did I commit this work (other than the obvious reasons). The talk is pretty simple & basically addressed to our Ph.D students, accessible to Master students and in which there is an introduction to my talk next week.